Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$

On a real analytic $5$-dimensional CR-generic submanifold $M^5 \subset \mathbb{C}^4$ of codimension $3$ hence of CR dimension $1$, which enjoys the generically satisfied nondegeneracy condition \begin{align*} {\bf 5} &= \text{rank}_\mathbb{C} \big( T^{1,0}M+T^{0,1}M + \big[T^{1,0}M,\,T^{0,1}M\big] \,+ \\&\qquad + \big[T^{1,0}M,\,[T^{1,0}M,T^{0,1}M]\big] + \big[T^{0,1}M,\,[T^{1,0}M,T^{0,1}M]\big] \big), \end{align*} a canonical Cartan connection is constructed after reduction to a certain partially explicit $\{ e\}$-structure of the concerned local biholomorphic equivalence problem.